Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 13 Dynamic Hedging

13.1 Introduction

In a friction-less market, dynamic delta hedging is a perfect method to hedge against price changes of a derivative instrument when the underlying of the option is the only source of risk, its price paths are continuous and volatility is constant. This is, for example, the case in the benchmark model of Black-Scholes-Merton (BSM, cf. Wilmott et al. (1995), ch. 3). In fact, it is one approach—another one being an equilibrium argument—to come up with the famous analytical formula of BSM. Independent of the particular model at hand, the delta of, for example, a put option P is defined by the first derivative of the option’s value with respect to the value of the underlying S

Delta hedging the put P then says that adding − Δ^P_t units of the underlying at time t to the put option completely neutralizes the price changes in the put option due to changes in the underlying. One then has for all t that

Investment banks are also often interested in replicating the payoff of such a put (or another option). This is accomplished by setting up a replication portfolio consisting of Δ^P_t units of the underlying and γ_t ≡ P_t − Δ^P_tS_t units of the risk-less bond B_t such that the resulting portfolio value equals the option value at any time t

or